Truth, Disjunction, and Induction

TL;DR

This paper investigates the limits of adding axioms to a theory of truth over Peano arithmetic, showing that even weak axioms like disjunctive correctness lead to non-conservativity and equate to adding induction, thus revealing the boundaries of truth theory extensions.

Contribution

It proves that adding disjunctive correctness to a truth theory over PA is non-conservative and equivalent to adding Δ₀-induction, extending previous results and developing a new proof technique.

Findings

Adding disjunctive correctness implies consistency of PA.

The theory with disjunctive correctness equals a theory with Δ₀-induction.

Develops a new proof method based on G"odel's second incompleteness theorem.

Abstract

By a well-known result of Kotlarski, Krajewski, and Lachlan (1981), first-order Peano arithmetic $PA$ can be conservatively extended to the theory $CT^{-}[PA]$ of a truth predicate satisfying compositional axioms, i.e., axioms stating that the truth predicate is correct on atomic formulae and commutes with all the propositional connectives and quantifiers. This results motivates the general question of determining natural axioms concerning the truth predicate that can be added to $CT^{-}[PA]$ while maintaining conservativity over $PA$. Our main result shows that conservativity fails even for the extension of $CT^{-}[PA]$ obtained by the seemingly weak axiom of disjunctive correctness $DC$ that asserts that the truth predicate commutes with disjunctions of arbitrary finite size. In particular, $CT^{-}[PA] + DC$ implies $Con(PA)$. Our main result states that the theory $CT^{-}[PA] + DC$ coincides with the theory $CT_0[PA]$ obtained by adding $\Delta_0$-induction in the language with the truth predicate. This result strengthens earlier work by Kotlarski (1986) and Cie\'sli\'nski (2010). For our proof we develop a new general form of Visser's theorem on non-existence of infinite descending chains of truth definitions and prove it by reduction to (L\"ob's version of) G\"odel's second incompleteness theorem, rather than by using the Visser-Yablo paradox, as in Visser's original proof (1989).